Open research questions in Point processes and geometric inequalities
103 gap statements mined from Point processes and geometric inequalities papers in our 4.5M-paper local library, which holds 550 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The Lp Minkowski problem remains unsolved for p < 1. The problem of characterizing geometric measures of convex bodies is fundamental and remains open.
The problem of finding a capillary convex body with a prescribed k-th capillary area measure is a challenging problem. The authors need to establish the existence and uniqueness of a smooth solution under a natural sufficient condition.
The capillary Christoffel-Minkowski problem is a new problem that has not been studied before. The authors identify a gap in the literature on the study of capillary convex bodies.
Comparison of spectra and open questions As we mentioned in the Introduction, there is a close relation between volumes of fully augmented links and volumes of ideal right-angled polyhedra. In conclusion, we formulate several open questions and one conjecture.
Normalized volume spectra of right-angled hyperbolic polyhedra · 2026The authors note that the capillary Orlicz-Minkowski problem has not been previously studied.
Future research can focus on developing more efficient algorithms for computing the arithmetic width. The arithmetic width can be used to study the behavior of lattice points in convex bodies. The results of the paper can be generalized to other types of convex bodies and geometric objects.
Despite the result obtained in this work there are still some open question such as: is it possible to find the optimal exponent in Theorems 1.2 and 1.3? Diaz and Weinstein established upper and lower bounds for the torsional rigidity of a beam, including in the case of multiply connected domains, in terms of its second-order momentum. It would be interesting to investigate a quantified version of this result.
Future research can focus on generalizing the method to other families of height functions. Future research can focus on applying the method to other areas of number theory.
The paper identifies a need for a method to find upper and lower bounds for the essential minimum.
The gap in prior work is that the inequality by Acosta-Duran can be refined. The paper identifies the need for a new Bonnesen-type inequality for the Poincaré constant in terms of diameter and inradius.
The study suggests that future research could focus on the practical applications of the results. The paper does not provide specific future research directions. The results may have implications for the study of convex bodies and their properties.
The research gap is focused on the theoretical aspects of convex geometry.
The third problem in Ben Green's collection of 100 open problems asks whether an open subset of $(0,1)$ that does not contain $x,y,z$ with $xy=z$ must have measure at most 1/3.
Product-free subsets of $(0,1)$ · 2026Future research could focus on applying the generalized Reifenberg theorem to specific subsets of R^n. Future research could focus on studying the properties of δ-regular maps. Future research could focus on generalizing the theorem to other types of subsets.
The isoperimetric problem in 2-dimensional spherically symmetric Randers space has not been previously studied using the Holmes-Thompson volume form. There is a lack of understanding of the geometry of Finsler spaces and the Randers metric.
Future research could focus on finding the precise Radon number for convex lattice sets in Z d for d ≥ 3. Future research could focus on generalizing the result to non-separable convexity spaces. Future research could focus on studying the complexity of halfspace collections.
The gap is that the fractional Helly number for convex lattice sets was known, but not for general separable convexity spaces. The gap is that the connection between the dual VC-dimension and the fractional Helly property was not well understood.
The paper suggests that future research should focus on developing new algorithms for convex optimization problems using the characterization of homogeneous convex cones with matching rank and Carathéodory number. The authors propose that future work should investigate the applications of the results in other areas of mathematics and computer science.
The authors note that prior work on convex conic programming and homogeneous convex cones has not fully characterized the Carathéodory number.
The multicover bifiltration is hard to compute, with the smallest known equivalent simplicial bifiltration having O(|X|^{d+1}) simplices. Prior approximate methods are exponential and only linear in |X| when taking k to be at most a constant.
Detecting gaps in the lattice point distribution. Developing algorithms for computing the arithmetic width. Establishing connections between the arithmetic width and discrete geometry, integer programming, and additive combinatorics.
Investigate the extension of Grünbaum's inequality to non-convex bodies - Explore the application of the Borell-Brascamp-Lieb inequality to other areas of mathematics - Study the properties of s-concave measures for s >…
To study the isoperimetric problem in other types of Finsler spaces or metrics. To determine the optimal shape that maximizes the area for a given perimeter in other geometric spaces.
The paper identifies a gap in the understanding of generalized convex bodies of revolution - The study aims to fill this gap by characterizing…
The curse of dimensionality makes convex hull computation challenging in high-dimensional spaces, - Exact algorithms are impractical for high-dimensional data, - Need for efficient approximation frameworks and modern computational…
Most-cited papers in Point processes and geometric inequalities
- A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points · Psychometrika · 1968 · 1,005 citations
- Discussion of a Set of Points in Terms of their Mutual Distances · Psychometrika · 1938 · 568 citations
- Convex hulls of finite sets of points in two and three dimensions · Communications of the ACM · 1977 · 521 citations
- A Survey of the Statistical Theory of Shape · Statistical Science · 1989 · 355 citations
- A note on a method for generating points uniformly on n -dimensional spheres · Communications of the ACM · 1959 · 308 citations
- The Convexity of Hilltops · The Journal of Geology · 1909 · 281 citations
- Partition of Space · American Mathematical Monthly · 1943 · 213 citations
- On Sets of Distances of n Points · American Mathematical Monthly · 1946 · 156 citations
- Computation of Minimum-Volume Covering Ellipsoids · Operations Research · 2004 · 138 citations
- The Theorems of Bony and Brezis on Flow-Invariant Sets · American Mathematical Monthly · 1972 · 138 citations
Most recent work
- Learned spatially varying microscopy model with adaptive point spread functions · Optics Express · 2026
- Sharp quantitative stability for the Minkowski first inequality via a quadratic estimate for the cone-volume measure · Advances in Mathematics · 2026
- Algèbres et volumes des polytopes · Journées mathématiques X-UPS · 2026
- GORENSTEIN FANO POLYTOPES ARISING FROM ORDER POLYTOPES AND CHAIN POLYTOPES · Journal of Commutative Algebra · 2026
- Nash Uniformization of Chessboard Sets by Nash Manifolds with Corners · ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE · 2026
- Bayesian inference for aggregated Hawkes processes · Bayesian Analysis · 2026
- What You See Is Not What Is There: Mechanisms, Models and Methods for Point Pattern Deviations · Statistical Science · 2026
- New Brunn–Minkowski and functional inequalities via convexity of entropy · Advances in Mathematics · 2026
- Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities · Advances in Mathematics · 2026
- A Klain–Schneider theorem for vector-valued valuations on convex functions · Journal of Functional Analysis · 2026
Find a gap in your own Point processes and geometric inequalities sub-topic
This page shows what the Point processes and geometric inequalities literature already flags as unresolved. To narrow it to your specific question, search the Research Gap Finder: the search is free with a free account and lists the papers closest to your topic first. Unlocking that topic (50 credits, charged once) fills the comparison table from our 4.5M-paper local library and writes the gaps from its rows.
Open the Research Gap Finder →